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In this article, a current control algorithm is conceived for converting solar dc power into ac power of high-quality. This conversion is based only on a dc-ac converter, avoiding the use of two stage converter. The proposed algorithm is founded on tracking the maximum power point which ensures controlling the electrical power exchanged with the grid. To assure a good dynamics and reduce the control errors, a proportional resonant controllers are utilized and a self-tuning founded on particle swarm optimization is utilized. The proposal strategy goodness is verified by means of numerical simulations.

The paper deals with a Second Order Variable Structure Control (VSC) for Wind Turbine PMSG-based and Generator-side Converter System. By means of the proposed control and the associated sliding observer, a vectorial control of PMSG generator is recognized while maximum power point tracking by the renewable source and DC voltage control of the generator-side converter are assured to maintain a high robustness in the face of parameters and load uncertainties within all the range variations of the controlled quantities.

The paper deals with a full sliding mode approach for induction motor drives direct torque and flux control. The proposed solution is composed by an integral sliding-mode control and by a speed-sensorless sliding mode observer. The suggested Integral Sliding-Mode Direct Torque Control (ISMDTC) is formulated in the stator flux reference frame, without employing current regulators. The adopted sliding mode observer provides stator-flux and torque estimation, without rotor-speed adaptation. The numerical results show that the ISMDTC maintains the robustness of the sliding controller, reducing the chattering during the steady state. Moreover, it consents to achieve similar performance like more complex second-order SMC offering more simple implementation. The ISMDTC assures good accuracy and robustness to parameters deviations at low-speed operations as well as in all the operative range of the IM drives. For proving the goodness of the proposal the suggested control is also compared with a conventional DTC control.

The impact of granular microstructure in permanent magnets on eddy current losses are investigated. A numerical homogenization procedure for electrical conductivity is defined. Then, an approximated simple analytical model for the homogenized conductivity able to capture the main features of the geometrical and material dependences is derived. Finally eddy current losses analytical calculations are given, and the two asymptotic expressions for losses in the stationary conduction limit and advanced skin effect limit are derived and discussed., 5 pages, 7 figures

In this paper a grid-tied photovoltaic (PV) cascaded H-bridge (CHB) inverter with embedded storage units is proposed. The multi-input single-output inverter architecture allows a distributed maximum power point tracking (DMPPT) approach, thus mitigating the mismatch effects. The energy storage units (ESUs) embedded at each dc-link provides more flexibility to the power circuit compensating for the inherent variability of the PV sources. The proposed power system ensures a continuous power supply to the grid also in case of PV power fluctuations due to irradiance conditions as well as to the MPPT. In particular, the presence of a storage element allows a voltage sweep on the PV source to be implemented, thus guaranteeing the maximization of PV energy harvesting, with no detrimental effect on the power delivery to the utility network. The numerical results, carried out on a single-phase nineteen-level PV CHB inverter, reveal the improved performance of the proposed design and control technique.

The study presents a significant development of an advanced control strategy for stand-alone three-phase induction generators. The system under study consists of a squirrel-cage induction generator driven mechanically by an internal combustion engine and electrically by a fault-tolerant converter coupled with a LC filter. As a novelty, in this study three low-cost electrical contactors are introduced in order to re-arrange the converter topology after a fault occurrence and a proper re-starting procedure is developed, allowing the converter to operate with only two legs. The control architecture is mainly described with respect to this last operating condition, evidencing its intrinsic robustness. The study focuses on the composite control of the converter and the prime mover in order to improve the dynamic assessment of the whole system. The converter control is designed around the sliding mode technique, due to its attractive performance, especially as far as parameter uncertainties are concerned. In order to validate the theoretical framework, this study presents the first ever made experimental tests of this system with reference to a generation unit of about 10 kVA. The results confirm the high capability of the proposed system in regulating voltage and frequency with very good dynamic features.

The full electric/hybrid electric propulsion is nowadays a promising technology for the aircraft. The problems related to the electric flight are inherent the storage system sizing, the design of lightweight motors and the power electronics converters. In this paper is proposed a preliminary study about the on-board wiring system. In particular, the maximum power transferrable with the common aeronautic wires in a distributed propulsion system has been investigated.

The paper firstly summarizes a simple analytical model of the air gap flux-density distribution for isotropic permanent magnet (PM) synchronous machines, in the presence of static eccentricity. The model was proposed by the authors in a previous paper and is based on an efficacious analytical expression of the variable length of air gap magnetic field lines which occur in eccentric brushless machines with surface-mounted permanent magnets. The approximate expression of the air gap field makes it possible to achieve a mathematical model with concentrated parameters close to that of a PM machine without eccentricity. The expression of the armature voltages and electromagnetic torque are found, also with reference to steady-state operating conditions at fixed rotor speed and impressed currents. The differences introduced by the considered type of eccentricity are evaluated and highlighted especially with reference to the air gap inductance and to waveforms and frequency spectra of voltages and shaft torque. Numerical results in a case-study of an 8-pole, 110 kW PM motor are compared to those obtained by using finite element analysis.

In this paper, an advanced control strategy for grid­tied photovoltaic (PV) cascaded H­bridge (CHB) inverter is proposed. The circuit topology consists of a proper number of power cells (H­bridge configuration) connected in series and supplied by individual PV modules. The adopted control method is a mixed staircase­PWM technique performed by means of a sorting algorithm to determine cells' switching state. The cells' state is related to the need of charging or discharging a particular cell much more than the others by calculating the voltage error at each dc­link (e.g., by considering the difference between the maximum power point tracking (MPPT) reference and the measured quantity). A dedicated PO thus, increasing the power extraction even in mismatched conditions. In order to prove the effectiveness and feasibility of the proposed approach, a set of experiments are performed on a laboratory prototype of a single­phase five­level PV CHB. The control section is implemented on FPGA by using a dSPACE real­time hardware platform; thus, obtaining fully dedicated digital circuits. Experimental results show good performance in terms of MPPT efficiency, total harmonic distortion, and power factor in both normal operation and mismatch conditions.

This study deals with direct torque control (DTC) of induction motor drives when a multilevel inverter is involved. The proposed control technique is finalised to reduce the amplitude of torque oscillations and to improve the motor dynamic response regardless of operating speed conditions while assuring a satisfactory input current distortion. This is achieved by using a special look-up-table built with a novel approach. The control itself is based on the application in each sampling interval of one or two voltage vectors suitably selected, in order to ensure low electromagnetic torque ripple in steady-state operations and fast dynamic behaviour in transient operations. In the algorithm, the converter voltage vectors selection is carried out by using a novel look-up table whose entry points are computed by means of a specialised relation, starting from the outputs of multilevel hysteresis controllers. The suggested method, suitable for a generic multilevel inverter, is detailed for a neutral point clamped three-level inverter. The proposed algorithm is validated by several numerical results, and compared with a DTC algorithm based on a traditional three-speed range look-up tables. Finally, some experimental results, obtained with a scale laboratory setup, are shown, discussed and compared with the numerical ones.

The paper deals with the control of three-phase voltage source rectifiers (VSRs) with an arbitrary number of voltage levels. By means of an approach borrowed from the “Direct Torque Control” of AC drives, the paper presents an enhanced direct power control (DPC) strategy for VSRs which directly controls both active (P) and reactive (Q) line power by considering the values of their time derivatives. This control strategy can be applied both to the Neutral Point Clamped and to Cascaded H-bridge multilevel converters. In every sampling interval the proposed control technique is able to select the most suitable converter voltage vector to obtain good dynamic response and to maintain P and Q within their respective reference bands. This result is achieved in the different operating conditions by means of a Dynamic Look-Up Table (D-LUT) of the converter voltage vectors, which are ordered on the basis of their influence on P and Q time derivatives. The mentioned D-LUT depends on the instantaneous operating point and is not predetermined. Some significant operating conditions are numerically investigated in order to show the capability of the modified DPC technique to ensure good values of power quality indexes for the grid. Experimental results on a laboratory setup based on a three-level VSR show good accordance with the simulation ones.

The paper suggests an energy management control strategy of wayside Li-ion capacitor (LiC) based energy storage for light railway vehicles (LRV). The installation of wayside supercapacitor (SC) storage devices, as widely recognized, allows the recovery of the braking energy for increasing the system efficiency as well as a better pantograph voltage profile. A new type of SC, LiC, interfaced with dc-interleaved converter has been presented. This technology has an energy density comparable to batteries and power density much higher than the batteries. The authors propose a control strategy based on the maximum kinetic energy recovery throughout braking operations of the running vehicles. The stored energy comes back to the vehicles during the accelerations. The strategy stays on the knowledge of the state of charge of LiC device and the actual vehicle speeds. In particular, the control algorithm evaluates, in real time, the actual value of LiC voltage and current references on the basis of the vehicles inertial forces and acceleration estimations, taking into account the power losses of the system. Experimental tests made on electromechanical simulator, equipped with a 136-V, 30.5-F LiC module, fully confirm the validity of the suggested control. Finally, experimental characterization of LiC module has been achieved.

Hybrid electric propulsion in the aviation field is becoming an effective alternative propulsion technology with potential advantages, including fuel savings, lower pollution, and reduced noise emission. On the one hand, the aeroengine manufacturers are working to improve fuel consumption and reduce pollutant emissions with new combustion systems, on the other hand, much attention is given to reducing the weight of the batteries increasing the energy density. Hybrid electric propulsion systems (HEPS) can take advantage of the synergy between two technologies by utilizing both internal combustion engines (ICEs) and electric motors (EMs) together, each operating at their respective optimum conditions. In the present work, some numerical investigations were carried out by using a zero-dimensional code able to simulate the flight mission of a turboprop aircraft, comparing fuel consumption and pollutant emissions of the original engine with other two smaller gas turbines working in hybrid configuration. An algorithm has been implemented to calculate the weight of the batteries for the different configurations examined, evaluating the feasibility of the hybrid propulsion system in terms of number of non-revenue passengers.

The paper deals with controlled induction motor for traction units of railways vehicles. Based on a conventional speed-sensorless vector-control of induction motor drives, some numerical investigations are carried out in order to verify the suitability of an open-loop observer in the context of railway traction applications with their specific requirements. In particular, the attention is focused on the control in the re-powering phase (e.g. after a coasting operation) and during an uphill start of the train with a steep slope; both of them usually are critical situations. Among the large amount of data available on-board of a modern railway vehicle, the propulsion control can use the speed signal usually delivered on the Ethernet bus in order to facilitate the re-synchronization of the inverter commands. Also the possible use of a low-accuracy speed detection is considered in order to improve the drive performance specifically in these operating conditions, where sensorless speed-identification is definitely incorrect. These speed estimations allow to improve the drive dynamic performance, avoiding at the same time the activation of protections against over-currents or over-voltages.

This work deals with an optimal control strategy for a three-phase grid-tied photovoltaic (PV) cascade H-bridge inverter. The main challenge for the control of this system topology derives from the inherent power imbalance among the phases as well as from the different power levels of the H-bridge cells in each phase leg. The power imbalance among the phases is compensated by the injection of a proper zero-sequence voltage. The power distribution among the cells of each phase could be affected by uneven irradiance and temperature conditions of PVGs (PV Generators). This latter issue is addressed by means of an adaptive modulation method especially suited to manage unequal dc sources, while assuring stable circuit operation and maximizing the power extraction through a dedicated MPPT (Maximum Power Point Tracking) algorithm. The digital controller is developed and tested in Matlab/Simulink environment integrated with XSG (Xilinx System Generator), thus allowing an easy transfer on FPGA embedded in dSPACE real-time platform. Simulations and experimental results prove the validity of the proposed design and control approach.

This paper deals with a Power Electronic Transformer (PET) topology for a 3-phase AC distribution grid. In the discussed topology, a Modular Multilevel Converter (MMC) and a Full-Bridge converter are employed for the medium voltage (MV) and the low voltage (LV) side, respectively. By using the space vector, approach a mathematical model for the MMC is presented and a grid-synchronous algorithm is implemented to easily control the power flow through the structure. The MV and LV side converters are linked through a High Frequency (HF) transformer, whose control strategy is a Dual-Active Phase-Shift Control (PSC) with Square Wave Modulation (SQM). This technique is combined with a predictive algorithm, which is able to keep each leg’s capacitors’ voltages balanced both in stationary and in transient conditions. The proposed algorithm is numerically validated in the Matlab/Simulink® environment.

This paper presents a control technique for smoothing voltage oscillations in the sub-modules of Modular Multilevel Converter during start-up operations of high-power medium-voltage induction motor drive. At steady-state, the power fluctuations in each sub-module at fundamental and 2 nd harmonic frequencies generate capacitor voltage oscillations which are much higher as the current frequency decreases and tend to infinite at zero frequency. This is a problem in variable speed drives, where very low frequency values are needed during start-up. The control injects a homopolar component of the converter output voltage and an alternative high frequency circulating current, whose magnitude is much higher than the induction motor one during start-up operations. A space vector modulation technique is used in the control implementation. A numerical investigation is carried out with reference to some relevant cases-study in order to show the validity of the proposed control which allows an effective reduction of voltage oscillations on the dc-link capacitors of the different converter sub-modules.

In this paper a measurement system for the characterization of electrical generators for the energy recovery from EMAs installed on airplanes is presented. The measurement system uses both mechanical transducers (position/speed/ - force) for the measure of mechanical power and electrical transducers (current - voltage) for the measure of electrical power. A test bench has been developed for the characterization and for the efficiency evaluation of the system for different values of speed, screw pitch and current supplied by the generator.

In this paper a power converter based on a modular multilevel topology is discussed. The Modular Multilevel Converter (MMC) is one of the emerging technology for high- or medium-voltage transformerless power conversion. The paper presents a space vector modulation technique developed for a MMC with arbitrary number of voltage levels; the presented technique integrates the modulation of the homopolar component of the converter output voltage. A numerical investigation is carried out with reference to some relevant cases-study in order to show the validity of the proposed modulation technique which allows an effective control of the converter recirculating currents.

In this paper an analysis of a power converter based on a modular multilevel topology is discussed. A mathematical model of the MMC is developed in order to evaluate its dynamic behavior and to better understand the incidence of the circulating current phenomenon which could compromise converter performance in variable speed drive applications. In fact, if such problem is not properly addressed reliability problems arise.

The paper presents power converter architectures for ultra-fast charging of road electric vehicles with the integration of distributed energy storage systems, which work as energy buffer between the grid and the electric vehicle in recharge. The study is introduced by an overview of the main issues related to the charging infrastructures in terms of electric vehicles charging times and their impact on the main grid. The paper continues with an analysis of ultra-fast charging infrastructures with particular focus on applications based on three-phase low voltage AC grid. Then, the proper design of modular multilevel converter architectures is presented taking into account its integration with energy storage modules of different technology, in order to reduce power requirements from the grid during the ultra-fast recharging phase. Different control strategies was proposed and simulated in terms of power flux control, during the charging and the discharging phase of the energy buffers, and voltage balancing among the different energy storage modules on the grid side.

In this paper a power converter based on a modular multilevel topology integrated with energy storage devices for directly supplying of traction motors is discussed. The Modular Multilevel Converter (MMC) is one of the emerging technology for high- or medium- voltage transformerless power conversion. The possibility to directly control the traction motors by means of a dc/ac MMC topology, using also integrated energy storage systems (ESS) for recovering the kinetic energy, is exploited. Furthermore, a simplified mathematical model of the MMC with integrated ESS has been developed in order to evaluate the behavior of the traction unit during a typical load-cycle. The control law of the traction motor, integrated with energy storage flow management, has been introduced. The numerical results reported in the paper permit to confirm the feasibility of the proposed design and control strategy for light travel vehicle.

With reference to anisotropic permanent magnet (PM) brushless motors, the papers deals with a procedure for on-line identification of electrical parameters, based on the “recursive least square method”, designed for the cases of strong parameters variation depending on the operating condition. The identification procedure is superimposed to a conventional control diagram of PM AC-brushless drives (either torque-control or speed-control) and is really very simple. The results of this analysis can be used as identifiers of failure conditions in the drive; otherwise, during motor characterization on a test-bench, the method can be a simple instrument to map motor parameters in different operating conditions. The proposed technique is tested by means of numerical investigation on a motor whose electrical parameters have been off-line evaluated by experimental measurements and mapped in different operating conditions.

A speed and high resolution position identification algorithm, suitable for brushless drives, is presented in this paper. The algorithm, which is developed basing upon the Popov's Hyperstability Theory, guarantees the convergence of the estimated signals to their corresponding actual values by means of the measured currents. The algorithm is proposed for brushless DC (BLDC) motors which are characterized by emfs shapes different from the ideal trapezoidal ones. The identification procedure can be performed starting from the knowledge of both low resolution rotor position Hall signals and emf shape. The identification algorithm is properly tested on a BLDC drive controlled by a predictive algorithm, by performing a simulation study in the Matlab-Simulink environment. The corresponding results have highlighted the effectiveness of the proposed sensorless control system at both low and high speed operation.

Electric propulsion represents the new research frontier in the future ship technology. There are different hybrid topologies requiring sizing criteria of components and related control strategies, finalized to maximize different objective functions: weight and space reduction on-board, fuel economy, pollution reduction or optimized efficiencies. In the paper sizing criteria of on-board sources and energy storage systems are proposed for a hybrid series architecture for a typical “passenger boat”. Two different design criteria are proposed for a series architecture: a criterion aims to ensure that the engine-generator group runs at fixed speed and power, whereas the other consists to run the engine at variable power and speed. The procedure for an optimal design of an on-board sources is set up and validated by numerical applications on a 23 m long passenger boat having a displacement of 37 tons.

The paper deals with isotropic PM-brushless drives in configuration “single-inverter, dual-motor” operating with unbalanced load-torques. After a short review of some control techniques based on the classical control methodologies of single-motor PM drives, an innovative control algorithm is presented. It is suitable to minimize the resultant armature current needed to obtain an assigned resultant motor torque, whatever is the load unbalance. Simplified analytical expressions are given in order to quickly evaluate optimized reference currents with good approximation. From these reference values, a predictive feeding algorithm evaluates the reference voltage space-vector for the inverter supplying the two motors in parallel. Current and torque oscillation, torque/current ratio, dynamic response and stability are the mainly observed quantities. Effectiveness of proposed control technique is highlighted.

An optimized control of PWM voltage source rectifiers (VSR) is presented. First, in a multi-input look-up table the different converter voltage-vectors are ordered with respect to their aptitude to increase (or decrease) the active power, depending on the angular position of the input voltage vector. A simple procedure allows the quick selection of the most suitable converter-voltage which ensures the correct slope sign of both active and reactive power, in order to satisfy the respective reference values. A predictive procedure evaluates the application time of the selected voltage vector within the actual sampling interval, in order to keep the instantaneous active power inside prefixed hysteresis bands. The proposed algorithm quickly evaluates also the voltage vector to be applied in the remaining part of the sampling interval. Numerical investigations in some heavy operating conditions are carried out at a fixed average value of the switching frequency, especially with reference to current distortion and asymmetry.

The paper deals with active voltage source rectifiers (PWM-VSR) having multilevel cascaded H-bridge configuration. With reference to operations with unsymmetrical supply voltages and/or unbalanced three-phase loads, some optimized control strategies are presented and compared. Particular attention is devoted to the most effective control algorithm in terms of distortion of the input-currents and limitation of the line-current unbalancing. The algorithms are represented by a set of algebraic equations able to generate in a very simple way the actual reference values for the dynamic control of VSRs. Some heavy transient operations are analyzed in order to show the validity of the proposed techniques and the capability to limit negative influence on the line power-quality indexes by means of optimized control of cascaded H-bridge active front-ends.

The paper deals with some problems concerning fault analysis in multilevel V.S.I. converters. With reference to cascaded H-bridge topology, first the possible conditions of "single fault" are classified by analyzing the modification of output voltages when a generic power device is damaged ("short-circuited" or "definitely open"). Then a procedure for on-line quick identification of the corrupted device is explained. It is based only on output voltages and currents measurements and does not require additional sensors. The minimum number of measurements allowing fault identification is linked to the number of converter levels. Finally, failure adapted techniques of converter modulation are presented and simulation results are discussed with reference to some significant cases. These techniques have the aim to minimize the influence of the different faults on the service of the whole drive.

In the paper a mathematical model of multilevel PWM-rectifiers is pointed out. The model formulation is generalized to "m" voltage levels and is unified for neutral-point-clamped, cascaded-H-bridge, and flying-capacitors topologies. A modulation technique is described for active front-ends. Specific solutions are proposed for the different regions of the complex plane correspondent to the m-level structure of the converter. Instead of complex look-up tables, very simple relations are given in order to identify the region and to select the most effective modulation pattern. Some opportunities to conveniently use the intrinsic redundancy of the converter topology are described.

The paper presents a novel control algorithm, for a stand alone generation plant equipped with induction machine. The aim of control is to regulate properly the voltage and frequency generation of induction machine during load power changes. A “decoupled” control technique has been proposed. The voltage and frequency have been controlled separately by means of prime torque and continuous variable capacitive impedance connected in parallel to generator clamps. Numerical results have given evidence of good performance of algorithm control in terms of steady state precision and time transient.

This paper presents a sensorless vector control system for induction motors by taking into account iron loss, in which a flux-observer-based method is applied. Since the flux observer is constructed in a synchronously rotating reference frame with respect to the rotor flux of a current model and the iron loss resistance of parallel exiting circuit is used, the proposed system is very simple and the compensation of iron loss related to the operating frequency is directly realized while calculating rotor fluxes and slip frequency. The accuracies of estimated torque and speed are improved. Furthermore, a high efficiency control method that minimizes the loss is proposed. The effectiveness of the proposed system has been verified by digital simulation and experimentation.

In high power railway traction applications the modulation of traction converters, at low switching carriers, produces harmonic currents in the range of audio frequencies. To ensure correct operation modes of the signalling circuits, the interference magnitudes levels have to be reduced. Psophometric current (p.c.) gives an index of the interference level produced by traction equipments: it takes into account the harmonic content of the primary current waveform in the multiwinding transformer. To limit the p.c. in the feeding network, a method for filtering harmonic currents produced by the modulation of interlaced PWM rectifiers (V.S.R.) is described. A properly sized additional winding is considered on the multicoil transformer used to supply the traction equipment. In the paper the system is analysed by means of the mathematical model and some design guidelines are proposed.

In railway traction applications the modulation of electrical traction drives in the range of radio frequencies produces currents harmonics. To ensure correct operation modes of the signaling circuits, the levels of interference magnitudes have to be reduced. Psophometric current (P.C.) gives an indication of the interference level produced by traction equipment: it takes into account the harmonic content of the waveform of primary current of the multibody traction transformer. To limit the P.C. a properly sized additional winding on the multicoil transformer used to supply the traction equipment is considered. In the paper the system is analyzed by means of the mathematical model and some design guidelines are proposed.

With reference to the traditional full-bridge converter topology, a very efficacious control technique of active front-ends (A.F.E.) is proposed. Centered on very fast dynamic predictive evaluation of some electrical quantities, this method allows a quick selection of the voltage space-vectors useful for the V.S. Rectifier modulation and the evaluation of the modulation intervals by means of very simple analytical expressions. The paper underlines the positive contribution of A.F.E. to the efficiency and power quality of systems feeding electrical drives.

The paper deals with the impact on the 3-wire feeding network of induction motor drives using direct torque control (DTC). Harmonic and interharmonic content of line currents is evaluated with reference to high-power drives (about 1 MW) with IGBT-voltage source inverter (VSI). The most common kinds of DTC algorithms are considered, and their performances both on motor-side and line-side are compared in a wide range of the motor speed. Results of numerical investigations highlight the different influence of the DTC methodologies on line current interharmonics at different motor speeds and give useful information for properly sizing the input filters.

The paper focuses on an energy management strategy for an electric vehicles powered with three power sources (PEM fuel cells, LiFePO4 battery and supercapacitors). The aim of the proposed strategy is to split the power demand between the different sources maximizing the driving range and the performances of the vehicle and increasing the life cycle of the battery and of the PEM fuel cell. To accomplish load sharing an energetic model of the overall system based on experimentally validated models of the PEM Fuel Cell, LiFePO4 battery and supercapacitors is formulated and an optimal control problem with inequality constraints is resolved using the particle swarm optimization technique (PSO) to search the global near-optimum at each sample interval. The results show that the proposed strategy guarantees energy saving and assures failsafe operations for the battery and the PEM Fuel Cell unaffecting the drivability of the vehicle and increasing autonomy.

The paper presents an analytical procedure for the evaluation of iron losses produced by eddy currents in the rotor sheets of axially laminated VR motors. These losses cannot be neglected especially in inverter fed closed loop operations. To simplify the eddy currents computation, the magnetic field in iron sheets is obtained by separately considering the contribution of the stator currents with removed rotor and the one of the Amperian currents, responsible for the rotor magnetisation. With reference to a generic nonsinusoidal steady-state condition, the proposed mathematical model gives the expression of the instantaneous values of the electromagnetic field and of the current density in each point of the rotor sheets. Iron losses are then obtained by means of a proper integration of the current density distribution. Finally, the paper shows the results of a numeric example which refers to a specific motor configuration and to an assigned waveshape of the input currents.

The paper tries to answer the question if and when unstable operations occur in practice in asynchronous motors supplied by variable frequency voltages. It is found and shown that unstable operating ranges are in practice greatly reduced in comparison with theoretical evaluations, obtained by simply considering the independent variation of motor electrical and geometrical parameters. The reduction is effectively due to the restriction of these ranges, which is connected to the physical relations among motor parameters which are taken into account by the designing techniques. It is also found that as greater is the output constant as smaller is the possibility to obtain red unstable operations. Illustrative tables are shown to summarize the results obtained and to give a quantitative answer to the first mentioned questions.